We settle a conjecture of Cerone and Dragomir on the concavity of the reciprocal of the Riemann zeta function on (1,∞). It is further shown in general that reciprocals of a family of zeta functions ...arising from semigroups of integers are also concave on (1,∞), thereby giving a positive answer to a question posed by Cerone and Dragomir on the existence of such zeta functions. As a consequence of our approach, weighted type Mertens sums over semigroups of integers are seen to be biased in favor of square-free integers with an odd number of prime factors. To strengthen the already known log-convexity property of Dirichlet series with positive coefficients, the geometric convexity of a large class of zeta functions is obtained and this in turn leads to generalizations of certain inequalities on the values of these functions due to Alzer, Cerone and Dragomir.
•A new method for setting the matrix parameter in the linearly involved GMC is proposed.•An alternative algorithm is presented to solve the linear involved convexity-preserving model.•Two properties ...of the solution path are proved to help with tuning parameter selection.
The generalized minimax concave (GMC) penalty is a newly proposed regularizer that can maintain the convexity of the objective function. This paper deals with signal recovery with the linearly involved GMC penalty. First, we propose a new method to set the matrix parameter in the penalty via solving a feasibility problem. The new method possesses appealing advantages over the existing method. Second, we recast the linearly involved GMC model as a saddle-point problem and use the primal-dual hybrid gradient (PDHG) algorithm to compute the solution. Another important work in this paper is that we provide guidance on the tuning parameter selection by proving desirable properties of the solution path. Finally, we apply the linearly involved GMC penalty to 1-D signal recovery and matrix regression. Numerical results show that the linearly involved GMC penalty can obtain better recovery performance and preserve the signal structure more successfully in comparison with the total variation (TV) regularizer.
Closest approach of a quantum projectile Kumar, A; Krisnanda, T; Arumugam, P ...
Journal of physics. Conference series,
05/2021, Volume:
1850, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
We consider the simplest case of Rutherford scattering, i.e. the head-on collision, where the projectile is treated quantum mechanically. The convexity of repulsive Coulomb force invokes a ...disagreement between the Ehrenfest’s and Hamilton’s dynamics. We show that the quantum projectile cannot approach as close as the corresponding classical one, and that the average distance of closest approach depends on the position spread of the wave function describing the projectile.
This study considers a periodic‐review joint pricing and inventory control problem for a single product, where production incurs a fixed cost plus a convex or concave variable cost. Our objective is ...to maximize the expected discounted profit over the entire planning horizon. We fully characterize the optimal policy for the single‐period problem. As the optimal policy for the multi‐period problem is too complicated to be implemented in practice, we develop well‐structured heuristic policies, and establish worst‐case performance bounds on the profit gap between the heuristic policies and the optimal policies. Numerical studies show that our heuristic policies perform extremely well. To further reveal the structural properties of the optimal policies, we also introduce two new concepts named κ‐convexity and sym‐κ‐convexity, provide the associated preservation results, and then characterize the optimal policies.
Some Topics on Convex Optimization Salman, Abbas Musleh; Alridha, Ahmed; Hussain, Ahmed Hadi
Journal of physics. Conference series,
03/2021, Volume:
1818, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
The aim of this paper is to clarify several important points, including a brief and adequate explanation of loving improvement, as well as laying out a number of important mathematical ...formulas that we need, supported with graphs.
In a triangle pathv1,…,vt of a graph G no edges exist joining vertices vi and vj such that |j−i|>2. A set S⊆V(G) is convex in the triangle path convexity of G, or t-convex, if the vertices of every ...triangle path joining two vertices of S are in S. The cardinality of a maximum proper t-convex subset of V(G) is the t-convexity number of G and the cardinality of a minimum set of vertices whose t-convex hull is V(G) is the t-hull number of G. We solve the fundamental complexity problems on the triangle path convexity. Among them, we present polynomial-time algorithms for determining the t-convexity number and the t-hull number of a general graph.
Let G be a graph, u and v two vertices of ... and ... a subset of ... A ... geodesic is a path between ... and ... of minimum length... is the set of vertices that lie on any ... geodesic and ... is ...the set ... is g-convex if ... The convexity number, ..., of ... is the maximum cardinality of a proper g-convex set of ... The clique number, , of ... is the maximum cardinality of a clique of ... If is a connected not complete graph then ... In this paper a necessary condition for ... is provided and, on the basis of this condition, both the class of distance-hereditary graphs for which ... and the class of chordal graphs for which ... are characterized.(ProQuest: ... denotes formulae omitted.)
Let G be a graph, u and v two vertices of G and X a subset of V(G). A u−vgeodesic is a path between u and v of minimum length. Ig(u,v) is the set of vertices that lie on any u−v geodesic and Ig(X) is ...the set ⋃u,v∈XIg(u,v). X is g-convex if Ig(X)=X. The convexity number, con(G), of G is the maximum cardinality of a proper g-convex set of G. The clique number, ω(G), of G is the maximum cardinality of a clique of G. If G is a connected not complete graph then ω(G)≤con(G). In this paper a necessary condition for ω(G)=con(G) is provided and, on the basis of this condition, both the class of distance-hereditary graphs for which ω(G)=con(G) and the class of chordal graphs for which ω(G)=con(G) are characterized.
The paper 'Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity' published in Proc. R. Soc. A 2013, 469, 20130075 (doi:10.1098/rspa.2013.0075) is ...clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. More precisely, if the quadratic form is Q*-convex then any solution of the Euler-Lagrange equations will necessarily minimize the integral. As a consequence, strict Q*-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of linear Euler-Lagrange equations in a given domain Ω with appropriate boundary conditions.
Semidefinite Programming (SDP) and Sums-of-Squares (SOS) relaxations have led to certifiably optimal non-minimal solvers for several robotics and computer vision problems. However, most non-minimal ...solvers rely on least squares formulations, and, as a result, are brittle against outliers. While a standard approach to regain robustness against outliers is to use robust cost functions, the latter typically introduce other non-convexities, preventing the use of existing non-minimal solvers. In this letter, we enable the simultaneous use of non-minimal solvers and robust estimation by providing a general-purpose approach for robust global estimation, which can be applied to any problem where a nonminimal solver is available for the outlier-free case. To this end, we leverage the Black-Rangarajan duality between robust estimation and outlier processes (which has been traditionally applied to early vision problems), and show that graduated non-convexity (GNC) can be used in conjunction with non-minimal solvers to compute robust solutions, without requiring an initial guess. we demonstrate the resulting robust non-minimal solvers in applications, including point cloud and mesh registration, pose graph optimization, and image-based object pose estimation (also called shape alignment). Our solvers are robust to 70-80% of outliers, outperform RANSAC, are more accurate than specialized local solvers, and faster than specialized global solvers. We also propose thefirst certifiably optimal non-minimal solver for shape alignment using SOS relaxation.
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